Tools

"... Well-structured transition systems provide the right foundation to compute a finite basis of the set of predecessors of the upward closure of a state. The dual problem, to compute a finite representation of the set of successors of the downward closure of a state, is harder: Until now, the theoretic ..."

Well-structured transition systems provide the right foundation to compute a finite basis of the set of predecessors of the upward closure of a state. The dual problem, to compute a finite representation of the set of successors of the downward closure of a state, is harder: Until now, the theoretical framework for manipulating downward-closed sets was missing. We answer this problem, using insights from domain theory (dcpos and ideal completions), from topology (sobrifications), and shed new light on the notion of adequate domains of limits.

"... Abstract. We describe a simple, conceptual forward analysis procedure for ∞-complete WSTS S. This computes the clover of a state s0, i.e., a finite description of the closure of the cover of s0. When S is the completion of a WSTS X, the clover in S is a finite description of the cover in X. We show ..."

Abstract. We describe a simple, conceptual forward analysis procedure for ∞-complete WSTS S. This computes the clover of a state s0, i.e., a finite description of the closure of the cover of s0. When S is the completion of a WSTS X, the clover in S is a finite description of the cover in X. We show that this applies exactly when X is an ω 2-WSTS, a new robust class of WSTS. We show that our procedure terminates in more cases than the generalized Karp-Miller procedure on extensions of Petri nets. We characterize the WSTS where our procedure terminates as those that are clover-flattable. Finally, we apply this to well-structured counter systems. 1

... and S is a continuous dcpo. Since S is a wpo, it is Noetherian in its Scott topology [25, Proposition 3.1]. Since S is a continuous dcpo, S is also sober [6, Proposition 7.2.27], so Corollary 6.5 of =-=[25]-=- applies: every closed subset F of S is such that MaxF is finite and F = ↓ MaxF . Now let F = Lub(CoverS(s0)). ⊓⊔ For any other representative, i.e., for any finite set R such that ↓ R = ↓ CloverS(s0)...

"... Many infinite state systems can be seen as well-structured transition systems (WSTS), i.e., systems equipped with a well-quasi-ordering on states that is also a simulation relation. WSTS are an attractive target for formal analysis because there exist generic algorithms that decide interesting veri ..."

Many infinite state systems can be seen as well-structured transition systems (WSTS), i.e., systems equipped with a well-quasi-ordering on states that is also a simulation relation. WSTS are an attractive target for formal analysis because there exist generic algorithms that decide interesting verification problems for this class. Among the most popular algorithms are acceleration-based forward analyses for computing the covering set. Termination of these algorithms can only be guaranteed for flattable WSTS. Yet, many WSTS of practical interest are not flattable and the question whether any given WSTS is flattable is itself undecidable. We therefore propose an analysis that computes the covering set and captures the essence of acceleration-based algorithms, but sacrifices precision for domain builds on the ideal completion of the well-quasi-ordered state space, and a widening operator that mimics acceleration and controls the loss of precision of the analysis. We present instances of our framework for various classes of WSTS. Our experience with a prototype implementation indicates that, despite the inherent precision loss, our analysis often computes the precise covering set of the analyzed system.

...r the more general clover algorithm [13]. These algorithms exploit the fact that every downward-closed subset of a well-quasi-ordering can be effectively represented as a finite union of order ideals =-=[12, 17]-=-. The covering set is then computed by identifying sequences of transitions in the system that correspond to loops leading from smaller to larger states in the ordering, and then computing the exact s...

"... Normally, one thinks of probabilistic transition systems as taking an initial probability distribution over the state space into a new probability distribution representing the system after a transition. We, however, take a dual view of Markov processes as transformers of bounded measurable function ..."

Normally, one thinks of probabilistic transition systems as taking an initial probability distribution over the state space into a new probability distribution representing the system after a transition. We, however, take a dual view of Markov processes as transformers of bounded measurable functions. This is very much in the same spirit as a “predicate-transformer ” view, which is dual to the state-transformer view of transition systems. We redevelop the theory of labelled Markov processes from this view point, in particular we explore approximation theory. We obtain three main results: (i) It is possible to define bisimulation on general measure spaces and show that it is an equivalence relation. The logical characterization of bisimulation can be done straightforwardly and generally. (ii) A new and flexible approach to approximation based on averaging can be given. This vastly generalizes and streamlines the idea of using conditional expectations to compute approximations. (iii) We show that there is a minimal process bisimulation-equivalent to a given process, and this minimal process is obtained as the limit of the finite approximants.

"... Abstract. Noetherian spaces are a topological concept that generalizes well quasiorderings. We explore applications to infinite-state verification problems, and show how this stimulated the search for infinite procedures à la Karp-Miller. 1 ..."

Abstract. Noetherian spaces are a topological concept that generalizes well quasiorderings. We explore applications to infinite-state verification problems, and show how this stimulated the search for infinite procedures à la Karp-Miller. 1

...etherian ring is Noetherian. My purpose is to stress the fact that Noetherian spaces are merely a topological generalization of the well-known concept of well quasi-orderings, a remark that I made in =-=[19]-=- for the first time. Until now, this led me into two avenues of research. The first avenue consists in adapting, in the most straightforward way, the theory of well-structured transition systems (WSTS...

"... Abstract. We give a constructive proof of Kruskal’s Tree Theorem— precisely, of a topological extension of it. The proof is in the style of a constructive proof of Higman’s Lemma due to Murthy and Russell (1990), and illuminates the role of regular expressions there. In the process, we discover an e ..."

Abstract. We give a constructive proof of Kruskal’s Tree Theorem— precisely, of a topological extension of it. The proof is in the style of a constructive proof of Higman’s Lemma due to Murthy and Russell (1990), and illuminates the role of regular expressions there. In the process, we discover an extension of Dershowitz ’ recursive path ordering to a form of cyclic terms which we call µ-terms. This all came from recent research on Noetherian spaces, and serves as a teaser for their theory. 1

"... Pure names are identifiers with no relation between them, except equality and inequality. In previous works we have extended P/T nets with the capability of creating and managing pure names, obtaining ν-APNs and proved that they are strictly well structured (WSTS), so that coverability and boundedn ..."

Pure names are identifiers with no relation between them, except equality and inequality. In previous works we have extended P/T nets with the capability of creating and managing pure names, obtaining ν-APNs and proved that they are strictly well structured (WSTS), so that coverability and boundedness are decidable. Here we use the framework recently developed by Finkel and Goubault-Larrecq for forward analysis for WSTS, in the case of ν-APNs, to compute the cover, that gives a good over approximation of the set of reachable markings. We prove that the least complete domain containing the set of markings is effectively representable. Moreover, we prove that in the completion we can compute least upper bounds of simple loops. Therefore, a forward Karp-Miller procedure that computes the cover is applicable. However, we prove that in general the cover is not computable, so that the procedure is non-terminating in general. As a corollary, we obtain the analogous result for Transfer Data nets and Data Nets. Finally, we show that a slight modification of the forward analysis yields decidability of a weak form of boundedness called width-boundedness.

...s. There it is proved that the least completion of X (that contains an adequate domain of limits, in the sense of [12]) is the so called ideal completion of X, or equivalently, the sobrification of X =-=[14]-=-. We will see here that the ideal completion of the set of markings can be effectively represented by mapping markings to the domain MS(MS(P)) of finite multisets of finite multisets of places. For th...

by
Fernando Rosa-Velardo, María Martos-Salgado, David de Frutos-Escrig
, 2001

"... Pure names are identifiers with no relation between them, except equality and inequality. In previous works we have extended P/T nets with the capability of creating and managing pure names, obtaining ν-PNs and proved that they are strictly well structured (WSTS), so that coverability and boundednes ..."

Pure names are identifiers with no relation between them, except equality and inequality. In previous works we have extended P/T nets with the capability of creating and managing pure names, obtaining ν-PNs and proved that they are strictly well structured (WSTS), so that coverability and boundedness are decidable. Here we use the framework recently developed by Finkel and Goubault-Larrecq for forward analysis for WSTS, in the case of ν-PNs, to compute the cover, that gives a good over approximation of the set of reachable markings. We prove that the least complete domain containing the set of markings is effectively representable. Moreover, we prove that in the completion we can compute least upper bounds of simple loops. Therefore, a forward Karp-Miller procedure that computes the cover is applicable. However, we prove that in general the cover is not computable, so that the procedure is non-terminating in general. As a corollary, we obtain the analogous result for Transfer Data nets and Data Nets. Finally, we show that a slight modification of the forward analysis yields decidability of a weak form of boundedness called width-boundedness, and identify a subclass of ν-PN that we call dw-bounded ν-PN, for which the cover is computable.

"... ABSTRACT. Well-structured transition systems provide the right foundation to compute a finite basis of the set of predecessors of the upward closure of a state. The dual problem, to compute a finite representation of the set of successors of the downward closure of a state, is harder: Until now, the ..."

ABSTRACT. Well-structured transition systems provide the right foundation to compute a finite basis of the set of predecessors of the upward closure of a state. The dual problem, to compute a finite representation of the set of successors of the downward closure of a state, is harder: Until now, the theoretical framework for manipulating downward-closed sets was missing. We answer this problem, using insights from domain theory (dcpos and ideal completions), from topology (sobrifications), and shed new light on the notion of adequate domains of limits. 1.

...lled adequate domains of limits. We discuss them in Section 3. For now, let us note that the second author also proposed to use another notion of completion in another context, known as sobrification =-=[15]-=-. We need to recap what this is about. A topological space X is always equipped with a specialization quasi-ordering, which we shall write ≤ again: x ≤ y if and only if any open subset containing x al...

...F = ↓ Max F . Now every two elements of Max F are incomparable, i.e., Max F is an antichain: since S is wpo, Max F is finite. Remark 3.6. Lemma 3.5 generalizes to Noetherian spaces, which extend wqos =-=[Gou07]-=-: every closed subset F of a sober Noetherian space S is of the form ↓ Max F , with Max F finite [Gou07, Corollary 6.5]. Wpos are sober, and every continuous dcpo is sober in its Scott topology [AJ94,...